(Thermo) Energy as Taylor expansion

[RadiumBlue]

Homework Statement

[/B]
I've attached a screenshot of the problem, which will probably provide much better context than my retelling. I'm having problems with parts f and g. The most relevant piece of information is:

"To get used to the process of Taylor expansions in two variables, first we will let [itex] V [/itex] be a constant. For the following three functions compute [itex] \Delta E [/itex] in terms of [itex] \Delta T[/itex] with [itex] V [/itex] constant.

1:[itex] E = \alpha V T^{17}[/itex] where [itex] \alpha [/itex] is a constant. "
...
etc.

Homework Equations

Taylor series
[itex]\Delta E = \frac{dE}{dT} \Delta T [/itex]

The Attempt at a Solution

My problem with this question is I'm not quite sure what it's asking/what answer it wants. Does it want just the first two terms of the taylor expansion for each equation, using V as a constant?

I solved part e this way:

Taylor expansion of E with respect to T:
[itex] E(T) = E(T_i) + \frac{dE}{dT} (T-T_i) ... [/itex]

Using only the linear term as the problem states, and subtracting E(T_i)

[itex] E(T) - E(T_i) = \frac{dE}{dT} (T-T_i) [/itex]

Substituting

[itex] \Delta E = \frac{dE}{dT} \Delta T [/itex]I don't know how to proceed for part F. Would it be this for the first equation?

[itex] \alpha V T^{17} = \alpha V (T_i)^{17} + 17\alpha V T^{16} (T-(T_i)) [/itex]

[itex] \alpha V T^{17} - \alpha V (T_i)^{17} = 17\alpha V T^{16} (\Delta T) [/itex]

and so forth? Or is it more complicated than that?

 

[anathema]

Hello there,

Thank you for sharing your work and thought process! It looks like you have a good understanding of the Taylor series expansion and how to apply it to this problem. For part f, you are correct in using the same approach as you did for part e. However, the only difference is that you will have two variables, V and T, instead of just T. So your final answer should have both V and T terms.

For part g, it seems like the problem is asking for the second derivative of E with respect to T, which is the coefficient of the quadratic term in the Taylor series expansion. So you will need to use the same approach but take the second derivative of E with respect to T instead of the first derivative.

I hope this helps! Let me know if you have any further questions or if you need any clarification on any of the steps. Good luck!